2011-05-31

Graphs and molecules - 1

Background

Mathematics - referred to as the queen of the sciences by that great Karl Gauss - comprises a large number of fields. It provides abstractions of varying sophistication, together with a vast body of more specific results. Several of them have applications in areas far beyond their origins.

One such that is (albeit indirectly) relevant to our current context is topology. Topology can be defined as the study of properties of objects that are invariant under continuous maps. Topology has evolved into a large field, whose major components include point-set topology, algebraic topology and geometric topology. Of these, point-set topology lays the foundation, defining most of the basic concepts. People who take a course in mathematics at the college level are familiar with several of them, usually in specific forms as applicable to real numbers, etc. In generalized forms, concepts such as continuity remain simple, while others such as compactness (every open cover should have a finite subcover) can be surprisingly subtle!

Curves

Let I be an interval [x, y] of real numbers, with x ≠ y. Let f:I → R2. Further, let f be a continuous map. We can call the range of f a curve. It is common to call f itself the curve.

Now, suppose that the restriction of f to [x, y) is injective (i.e., f(a) = f(b) ⇒ a = b ∀ a, b ∈ [x, y)). The curve does not intersect itself, since the map is injective except for the single point y. Further suppose that f(x) = f(y). Such a curve is called a simple closed curve or a Jordan Curve.

Some texts define I above to be the interval [0, 1]. Another commonly found definition is: a Jordan Curve in R2 is one for which it is possible to construct an injective continuous map g:S1 → R2, where S1 is a circle. All the definitions are equivalent. Exercise: how? :-)

Graphs

Evidently, when we move to a discrete scenario, concepts such as continuity have to be adapted appropriately. Also, the discrete scenario brings two distinct kinds of entities into the discussion: nodes (or vertices) and edges. Note that the same notions exist in geometry as well. For instance, a triangle has three nodes and three edges. A graph is a collection of nodes and edges, where each node has at least one edge (i.e., one adjacent node).

A path can be defined as the sequence vi of vertices, where an edge ei exists between the nodes vi and vi+1. The index i varies in the discrete interval [1, n], for an applicable value of n.

Analogous to the continuous scenario, we can define a simple closed path or a simple cycle as a path with no repeated nodes except for the starting/ending node.

For our immediate purposes, we are mostly interested in undirected graphs.

Connectedness, terminal nodes

Let us denote the nodes of a graph by N, and its edges by E. For x, y ∈ N, x ≠ y, if there exists a path from x to y, we say that x and y are connected. If the above holds good ∀ x, y ∈ N, then we say that the graph itself is connected.

Suppose that the graph is not connected. Then it contains two or more connected subgraphs. Obviously, they are pair-wise disjoint, i.e., any given node belongs to one and exactly one such subgraph. Similarly, any given edge belongs to one and exactly one such subgraph.

A node with only one adjacent node connected to it (even though it can have any number of edges to that single adjacent node) is called a terminal node. As a special case, we note that a node that is part of a simple closed path cannot be a terminal node. Exercise: why? :-)